Difference between revisions of "Dedekind cut"
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| − | A subdivision of the set of real (or only of the rational) numbers (of) $\mathbf R$ into two non-empty sets $A$ and $B$ whose union is $\mathbf R$, such that $a<b$ for every $a\in A$ and $b\in B$. A Dedekind cut is denoted by the symbol $A|B$. The set $A$ is called the lower class, while the set $B$ is called the upper class of $A|B$. Dedekind cuts of the set of rational numbers are used in the construction of the theory of real numbers (cf. [[Real number|Real number]]). The | + | A subdivision of the set of real (or only of the rational) numbers (of) $\mathbf R$ into two non-empty sets $A$ and $B$ whose union is $\mathbf R$, such that $a<b$ for every $a\in A$ and $b\in B$. A Dedekind cut is denoted by the symbol $A|B$. The set $A$ is called the lower class, while the set $B$ is called the upper class of $A|B$. Dedekind cuts of the set of rational numbers are used in the construction of the theory of real numbers (cf. [[Real number|Real number]]). The [[continuity axiom]] for the real line can be formulated in terms of Dedekind cuts of real numbers. |
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More generally we may define a cut in any [[totally ordered set]] $X$ to be a partition of $X$ into two non-empty sets $A$ and $B$ whose union is $X$, such that $a<b$ for every $a\in A$ and $b\in B$: a Dedekind cut is a cut in which either $A$ has a maximal element or $B$ has a minimal element. A [[continuous set]] is a totally ordered set in which all cuts are Dedekind cuts. | More generally we may define a cut in any [[totally ordered set]] $X$ to be a partition of $X$ into two non-empty sets $A$ and $B$ whose union is $X$, such that $a<b$ for every $a\in A$ and $b\in B$: a Dedekind cut is a cut in which either $A$ has a maximal element or $B$ has a minimal element. A [[continuous set]] is a totally ordered set in which all cuts are Dedekind cuts. | ||
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| + | See also: [[Completion, MacNeille (of a partially ordered set)|Dedekind–MacNeille completion]]. | ||
[[Category:Order, lattices, ordered algebraic structures]] | [[Category:Order, lattices, ordered algebraic structures]] | ||
Latest revision as of 20:56, 28 September 2016
cut
A subdivision of the set of real (or only of the rational) numbers (of) $\mathbf R$ into two non-empty sets $A$ and $B$ whose union is $\mathbf R$, such that $a<b$ for every $a\in A$ and $b\in B$. A Dedekind cut is denoted by the symbol $A|B$. The set $A$ is called the lower class, while the set $B$ is called the upper class of $A|B$. Dedekind cuts of the set of rational numbers are used in the construction of the theory of real numbers (cf. Real number). The continuity axiom for the real line can be formulated in terms of Dedekind cuts of real numbers.
Comments
For the construction of $\mathbf R$ from $\mathbf Q$ using cuts see [a1].
References
| [a1] | W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953) |
Comments
More generally we may define a cut in any totally ordered set $X$ to be a partition of $X$ into two non-empty sets $A$ and $B$ whose union is $X$, such that $a<b$ for every $a\in A$ and $b\in B$: a Dedekind cut is a cut in which either $A$ has a maximal element or $B$ has a minimal element. A continuous set is a totally ordered set in which all cuts are Dedekind cuts.
See also: Dedekind–MacNeille completion.
Dedekind cut. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dedekind_cut&oldid=33795